Minimum Equitable Dominating Randi´c Energy of a Graph

International J.Math. Combin. Vol.1(2018), 97-108

Minimum Equitable Dominating Randić Energy of a Graph Rajendra P. (Bharathi College, PG and Research Centre, Bharathinagara, 571 422, India)

R.Rangarajan (DOS in Mathematics, University of Mysore, Mysuru, 570 006, India)

prajumaths@gmail.com, rajra63@gmail.com

Abstract: Let G be a graph with vertex set V (G), edge set E(G) and di is the degree of its i-th vertex vi , then the Randić matrix R(G) of G is the square matrix of order n, whose (i, j)-entry is equal to √ 1 if the i-th vertex vi and j-th vertex vj of G are adjacent, di dj

and zero otherwise. The Randic energy [3] RE(G) of the graph G is defined as the sum of the absolute values of the eigenvalues of the Randić matrix R(G). A subset ED of V (G) is called an equitable dominating set [11], if for every vi ∈ V (G) − ED there exists a vertex

vj ∈ ED such that vi vj ∈ E(G) and |di (vi ) − dj (vj )| ≤ 1. In the contrast, such a dominating

set ED is Smarandachely if |di (vi ) − dj (vj )| ≥ 1. Recently, Adiga, et.al. introduced, the

minimum covering energy Ec(G) of a graph [1] and S. Burcu Bozkurt, et.al. introduced, Randić Matrix and Randić Energy of a graph [3]. Motivated by these papers, Minimum equitable dominating Randić energy of a graph REED (G) of some graphs are worked out and bounds on REED (G) are obtained.

Key Words: Randić matrix and its energy, minimum equitable dominating set and minimum equitable dominating Randić energy of a graph.

AMS(2010): 05C50. §1. Introduction Let G be a graph with vertex set V (G) and edge set E(G). The adjacency matrix A(G) of the graph G is a square matrix, whose (i, j)-entry is equal to 1 if the vertices vi and vj are adjacent, otherwise zero [12]. Since A(G) is symmetric, its eigenvalues are all real. Denote them by λ1 , λ2 , . . . , λn , and as a whole, they are called the spectrum of G and denoted by Spec(G).

The energy of graph [12] G is ε(G) =

Pn

i=1

|λi |.

The literature on energy of a graph and its bounds can refer [4,8,9,10,12]. The Randić matrix R(G) = (rij ) of G is the square matrix of order n, where 1 Received

July 28, 2017, Accepted March 1, 2018.

98

Rajendra P. and R.Rangarajan

(rij ) =

  √1

di dj

 0,

, if vi and vj are adjacent vertices in G; otherwise.

The Randic energy [3] RE(G) of the graph G is defined as the sum of the absolute values of the eigenvalues of the Randić matrix R(G). Let ρ1 , ρ2 , · · · , ρn be the eigenvalues of the Randić matrix R(G). Since R(G) is symmetric matrix, these eigenvalues are real numbers and their sum is zero. Randić energy [3] can be defined as Pn RE(G) = i=1 |ρi |

For details of Randić energy and its bounds, can refer [2, 3, 5, 6, 7].

Let G be a graph with vertex set V (G) = {v1 , v2 , · · · , vn } and ED is minimum equitable dominating set of G. Minimum equitable dominating Randić matrix of G is n × n matrix RED (G) = (rij ), where  1    √di dj , if vi and vj are adjacent vertices in G;  (rij ) = 1, if i = j and vi ∈ ED;     0, otherwise.

The characteristic polynomial of RED (G) is denoted by det(ρI−RED (G)) = |ρI−RED (G)|.

Since RED (G) is symmetric, its eigenvalues are real numbers. If the distinct eigenvalues of RED (G) are ρ1 > ρ2 > · · · > ρr with their multiplicities are m1 , m2 , · · · , mr then spectrum of RED (G) is denoted by   ρ1 ρ2 · · · ρr .

SpecRED (G) =  m1 m2 · · · mr The minimum equitable dominating Randić energy of G is defined as REED (G) =

n X i=1

|ρi |.

Example 1.1 Let W5 be a wheel graph, with vertex set V (G) = {v1 , v2 , v3 , v4 , v5 }, and let its minimum equitable dominating set be ED = {v1 }. Then minimum equitable dominating Randić matrix RED (W5 ) is 

1

  √1  12  1 RED (W5 ) =   √12  1 √  12 √1 12



√1 12

√1 12

0

√1 12 1 3

1 3

0

1 3

0

1 3

0

1 3

0

1 3

0



√1 12  1  3 

 0    1  3  0

 −0.6666 0 0.2324 1.4342 .

SpecRED (W5 ) =  1 2 1 1

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Minimum Equitable Dominating Randić Energy of a Graph

The minimum equitable dominating Randić energy of W5 is REED (W5 ) = 2.3332.

§2. Bounds for the Minimum Equitable Dominating Randić Energy of a Graph Lemma 2.1 If ρ1 , ρ2 , · · · , ρn are the eigenvalues of RED (G). Then n X i=1

and

n X i=1

ρi = |ED|

ρ2i = |ED| + 2

X 1 , dd i<j i j

where ED is minimum equitable dominating set.

Proof (i) The sum of eigenvalues of RED (G) is n X

ρi =

i=1

n X i=1

rii = |ED|.

(ii) Consider, the sum of squares of ρ1 , ρ2 , . . . , ρn is, n X

ρ2i

=

n X n X i=1 j=1

i=1

=

n X i=1

n X i=1

rij rji =

ρ2i

n X

(rii )2 +

i=1

(rii )2 + 2

X

rij rji

i6=j

X (rij )2 i<j

X 1 = |ED| + 2 . dd i<j i j

2

Upper and lower bounds for REED (G) is similar proof to McClelland’s inequalities [10], are given below Theorem 2.2(Upper Bound) Let G be a graph with ED is minimum equitable dominating set.

Then v   u u X 1  u REED (G) ≤ tn |ED| + 2 . d dj i i<j Proof Let ρ1 , ρ2 , · · · , ρn be the eigenvalues of RED (G). By Cauchy-Schwartz inequality, we have !2 ! n ! n n X X X b2i , (1) ai b i ≤ a2i i=1

where a and b are any real numbers.

i=1

i=1

100

Rajendra P. and R.Rangarajan

If ai = 1, bi = |ρi | in (1), we get n X i=1

!2

|ρi |

≤

n X

12

i=1

by Lemma 2.1,

!

n X

|ρi |2

i=1



 X 1  , [REED (G)]2 ≤ n |ED| + 2 d d i j i<j

!

v   u u X 1  u REED (G) ≤ tn |ED| + 2 . d d i<j i j

Theorem 2.3(Lower Bound) Let G be a graph with |ED| is minimum equitable dominating set and di is degree of vi . Then REED (G) ≥ where D =

Qn

i=1

s

|ED| + 2

X 1 2 + n(n − 1)D n , dd i<j i j

|ρi |.

Proof Consider 2

[REED (G)] =

"

n X i=1

#2

|ρi |

=

n X i=1

|ρi |2 +

X i6=j

|ρi | |ρj |.

(2)

By using arithmetic and geometric mean inequality, we have 1 n(n − 1)

X i6=j

X i6=j

X i6=j

|ρi | |ρj | |ρi | |ρj | |ρi | |ρj |

≥ ≥ ≥

1   n(n−1) Y  |ρi | |ρj |

i6=j

n(n − 1) n(n − 1)

n Y

|ρi |

n Y

|ρi |

i=1

i=1

2(n−1)

!2/n

1 ! n(n−1)

.

Now using (3) in (2), we get 2

[REED (G)] ≥

n X i=1

2

|ρi | + n(n − 1)

[REED (G)]2 ≥ |ED| + 2 REED (G) ≥

s

|ED| + 2

n Y

i=1

!2/n

|ρi |

,

n Y X 1 2 + n(n − 1)D n , where D = |ρi |, dd i<j i j i=1

X 1 2 + n(n − 1)D n . d d i j i<j

(3)

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Minimum Equitable Dominating Randić Energy of a Graph

§3. Bounds for Largest Eigenvalue of RED (G) and its Energy

Proposition 3.1 Let G be a graph and ρ1 (G) = max1≤i≤n {|ρi |} be the largest eigenvalue of RED (G). Then v   u s u X 1 X 1 u1   t |ED| + 2 ≤ ρ1 (G) ≤ |ED| + 2 . n dd dd i<j i j i<j i j Proof Consider, ρ21 (G) ρ1 (G)

=

max1≤i≤n {|ρi |2 } ≤

≤

s

|ED| + 2

X 1 dd i<j i j

n X i=1

|ρi |2 = |ED| + 2

X 1 dd i<j i j

Next, n ρ21 (G) ≥ ρ21 (G) ≥

ρ1 (G) ≥ Therefore,

n X

X 1 dd i=1 i<j i j   X 1 1  |ED| + 2 n d d i j i<j v   u u X 1 u1   t |ED| + 2 n d d i j i<j ρ2i = |ED| + 2

v   u s u X X 1 1  u1  t |ED| + 2 ≤ ρ1 (G) ≤ |ED| + 2 . n dd dd i<j i j i<j i j

Proposition 3.2 If G is a graph and n ≤ |ED| + 2 REED (G) ≤

|ED| + 2

P

1 i<j di dj

n

P

1 i<j di dj ,

v  u u X u + t(n − 1) |ED| + 2 i<j

then

1 − di dj

|ED| + 2

P

n

1 i<j di dj

!2  .

Proof We know that, n X i=1

ρ2i = |ED| + 2

X 1 , dd i<j i j

n X i=2

ρ2i = |ED| + 2

X 1 − ρ21 d d i j i<j

(4)

102

Rajendra P. and R.Rangarajan

By Cauchy-Schwarz inequality, we have n X

ai b i

i=2

!2

≤

n X

a2i

i=2

!

n X

b2i

i=2

!

.

If ai = 1 and bi = |ρi |, we have n X i=2

!2

|ρi |

≤

2

≤

REED (G)

≤

[REED (G) − |ρ1 |]

(n − 1)

n X i=2

|ρi |2



 X 1 (n − 1) |ED| + 2 − ρ21  d d i j i<j v   u u X 1 u ρ1 + t(n − 1) |ED| + 2 − ρ21  d d i j i<j

Consider the function, v   u u X 1 u F (x) = x + t(n − 1) |ED| + 2 − x2 . d d i j i<j Then,

p x (n − 1) F (x) = 1 − q .

P |ED| + 2 i<j di1dj − x2 ′

Here F (x) is decreasing in

  X X 1  1   1 |ED| + 2 , |ED| + 2 . n d d d dj i j i i<j i<j 

We know that F ′ (x) ≤ 0,

We have



p x (n − 1) 1− q ≤ 0.

P |ED| + 2 i<j di1dj − x2 x≥

Since,

s

|ED| + 2

P

n

1 i<j di dj

.

P X 1 |ED| + 2 i<j n ≤ |ED| + 2 and dd n i<j i j

1 di dj

≤ ρ1 ,

(5)

103

Minimum Equitable Dominating Randić Energy of a Graph

we have s

|ED| + 2

P

1 i<j di dj

n

≤

|ED| + 2

P

1 i<j di dj

n

≤ ρ1 ≤ |ED| + 2

X 1 . dd i<j i j

Then, equation (5) become

REED (G) ≤

|ED| + 2

P

n

1 i<j di dj

v  u u X 1 u + t(n − 1) |ED| + 2 − dd i<j i j

|ED| + 2

P

1 i<j di dj

n

!2  .

2

§4. Minimum Equitable Dominating Randić Energy of Some Graphs Theorem 4.1 If Kn is complete graph with n vertices, then minimum equitable dominating Randić energy of Kn is 3n − 5 REED (Kn ) = . n−1 Proof Let Kn be the complete graph with vertex set V (G) = {v1 , v2 , . . . , vn } and minimum equitable dominating set is ED = {v1 }, we have characteristic polynomial of RED (Kn ) is

ρ − 1

−1

n−1

−1

n−1

. |ρI − RED (Kn )| =

..

−1

n−1

−1

n−1

−1

n−1

−1 n−1

−1 n−1 −1 n−1

···

.. .

ρ .. .

··· .. .

−1 n−1 −1 n−1 −1 n−1

−1 n−1 −1 n−1 −1 n−1

···

ρ −1 n−1

···

··· ···

−1 n−1 −1 n−1 −1 n−1

.. .

−1 n−1 −1 n−1 −1 n−1

.. .

ρ

−1 n−1

−1 n−1 −1 n−1

−1 n−1

ρ

−1

n−1

−1

n−1

−1

n−1

..

.

−1

n−1

−1

n−1

ρ

.

n×n

1 Rk′ = Rk − R2 , k = 3, 4, · · · , n − 1, n. Then, we get ρ + n−1 common from R3 to Rn and we have

ρ − 1 −1 −1 −1 −1 −1

· · ·

n−1 n−1 n−1 n−1 n−1

−1 −1 −1 −1

−1 ρ · · · n−1 n−1 n−1

n−1

n−1

0

−1 1 · · · 0 0 0

n−2

.

1 . . . . . .

. . . . . . . . |ρI − RED (Kn )| = ρ + . . . . . .

. n−1

0 −1 0 ··· 1 0 0

0 −1 0 ··· 0 1 0

0 −1 0 ··· 0 0 1

n×n

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Rajendra P. and R.Rangarajan

C2′ = C2 + C3 + · · · + Cn , we get the characteristic polynomial |ρI − RED (Kn )| = SpecRED (Kn ) =

n−2

1 2n − 3 n−3 2 ρ+ ρ − ρ+ , n−1 n−1 n−1   √ √ 

−1 n−1

(2n−3)− 4n−3 2(n−1)

n−2

1

(2n−3)+ 4n−3 2(n−1) 

1

.

The minimum equitable dominating Randić energy of Kn is REED (Kn ) =

3n − 5 . n−1

2

Theorem 4.2 If Sn , (n ≥ 4) is star graph with n vertices, then minimum equitable dominating Randić energy of Sn is REED (Sn ) = n.

Proof Let Sn , (n ≥ 4) be the star graph with vertex set V (G) = {v1 , v2 , · · · , vn } and minimum equitable dominating set is ED = {v1 , v2 , · · · , vn }, we have characteristic polynomial of RED (Sn ) is

ρ − 1

√−1

n−1

√−1

n−1

. |ρI − RED (Sn )| = ..

−1

√

n−1

−1

√n−1

−1

√n−1

√−1 n−1

√−1 n−1

···

√−1 n−1

√−1 n−1

ρ−1

0

···

0

0

0 .. .

ρ−1 .. .

··· .. .

0 .. .

0 .. .

0

0

···

ρ−1

0

0

0

···

0

ρ−1

0

0

···

0

0

√−1

n−1

0

0

..

.

0

0

ρ − 1

.

n×n

Rk′ = Rk − R2 , k = 3, 4, · · · , n. Then taking (ρ − 1) common from R3 to Rn , we get

ρ − 1

√−1

n−1

0

. n−2

|ρI − RED (Sn )| = (ρ − 1)

..

0

0

0

√−1 n−1

√−1 n−1

···

√−1 n−1

√−1 n−1

ρ−1

0

···

0

0

−1 .. .

1 .. .

··· .. .

0 .. .

0 .. .

−1

0

···

1

0

−1

0

···

0

1

0

0

−1

0

···

C2′ = C2 + C3 + · · · + Cn . Then, the characteristic polynomial |ρI − RED (Sn )| = ρ(ρ − 1)n−2 (ρ − 2),

√−1

n−1

0

0

..

.

0

0

1

.

n×n

105

Minimum Equitable Dominating Randić Energy of a Graph



 SpecRED (Sn ) =  

0

 2  .  1

1

1 n−2

2

The minimum equitable dominating Randić energy of Sn is REED (Sn ) = n.

Theorem 4.3 If Km,n , where m < n and |m − n| ≥ 2 is complete bipartite graph with m + n vertices, then minimum equitable dominating Randić energy of Km,n is REED (Km,n ) = m + n.

Proof Let Km,n , where m < n and |m − n| ≥ 2 be the complete bipartite graph with vertex set V (G) = {v1 , v2 , . . . , vm , u1 , u2 , · · · , un } and minimum equitable dominating set is ED = V (G), we have characteristic polynomial of RED (Km,n ) is

0

ρ − 1

ρ−1

0

..

..

. .

0 0

0 0 |ρI − RED (Km,n )| =

−1 −1

√mn √mn

√−1

mn √−1 mn

.. ..

. .

−1

√ √−1 mn

mn

−1

√mn √−1 mn

√−1 mn −1 √ mn

√−1 mn −1 √ mn

···

√−1 mn √−1 mn

√−1 mn √−1 mn

···

√−1 mn √−1 mn

ρ−1

0

···

0 .. .

ρ−1 .. .

··· .. .

√−1 mn √−1 mn

0

0

···

0

0

···

...

0

0

... .. .

0 .. .

0 .. .

··· ρ− 1 ···

0 ρ−1

0 √−1 mn √−1 mn

··· ··· .. .

.. .

.. .

√−1 mn √−1 mn

··· ···

.. .

.. .

√−1 mn −1 √ mn

··· .. .

.. .

√−1

mn

√−1

mn

..

.

−1 −1 √ √

mn mn

√−1 √−1

mn mn

.

0 0

0 0

.. ..

. .

ρ−1 0

0 ρ − 1

...

Rk′ = Rk − Rm , k = 1, 2, 3, · · · , m − 1 and Rd′ = Rd − Rm+1 , d = m + 2, m + 3, · · · , m + n.

Then, taking (ρ − 1) common from R1 to Rm−1 and Rm+2 to Rm+n , we get

1

0

..

.

0

m+n−2

0 (ρ − 1)

√−1

mn

0

..

.

0

0

0

···

0

−1

1 .. .

··· .. .

0 .. .

−1 .. .

0

···

1

−1

0 √−1 mn

···

0

0

0 .. .

0 .. .

··· .. . ···

0

0

√−1 mn

√−1 mn

√−1 mn

ρ−1

ρ−1

···

0

0 0 .. .

0

···

√−1 mn

0

···

0

···

√−1 mn

0 .. .

··· .. .

0 .. .

0 .. .

−1 .. .

1 .. .

··· .. .

0 .. .

0

···

0

0

−1

0

···

1

0

0

0

···

−1

0

···

0

0

0

..

.

0

√−1

mn

.

0

0

..

.

0

1

106

Rajendra P. and R.Rangarajan

′ Cm+1 = Cm+1 + Cm+2 + · · · + Cm+n ,

1

0

..

.

0

m+n−2

0 (ρ − 1)

√−1

mn

0

..

.

0

0

0

···

0

−1

0

0

···

0

1 .. .

··· .. .

0 .. .

−1 .. .

0 .. .

0 .. .

··· .. .

0 .. .

0

···

1

−1

···

···

0

0

0

0

√−n mn

√−1 mn

√−1 mn

···

√−1 mn

ρ−1

0

···

0

ρ−1

0

√−1 mn

···

√−1 mn

0 .. .

··· .. .

0 .. .

0 .. .

0 .. .

1 .. .

··· .. .

0 .. .

0

···

0

0

0

0

···

1

0

···

0

0

0

0

···

0

The characteristic polynomial |ρI − RED (Km,n )| = ρ (ρ − 1)m+n−2 (ρ − 2),  0 SpecRED (Km,n ) =  1

1

2

m+n−2 1

0

0

..

.

0

√−1

mn

.

0

0

..

.

0

1



.

The minimum equitable dominating Randić energy of Km,n is REED (Km,n ) = m + n.

2

Theorem 4.4 If Kn×2 , (n ≥ 3) is cocktail party graph with 2n vertices, then minimum equitable dominating Randić energy of Kn×2 is 4n−6 n−1 .

Proof Let Kn×2 , (n ≥ 3) be the cocktail party graph with vertex set V (G) = {v1 , v2 , · · · , vn , u1 , u2 , · · · , un } and minimum equitable dominating set is ED = {v1 , u1 }, we have characteristic polynomial of RED (Kn×2 ) is

λ − 1

0

−1

2n−2

−1

2n−2

.

.

. |ρI − RED (Kn×2 )| =

−1

2n−2

−1

2n−2

.

..

−1

2n−2

−1

2n−2

0

−1 2n−2

−1 2n−2

λ−1

−1 2n−2

−1 2n−2

−1 2n−2

λ

−1 2n−2

−1 2n−2

−1 2n−2

.. .

.. .

λ .. .

−1 2n−2

−1 2n−2

−1 2n−2

−1 2n−2

−1 2n−2

.. .

0 .. .

−1 2n−2

−1 2n−2

−1 2n−2

−1 2n−2

−1 2n−2

−1 2n−2

.. .

−1 2n−2

−1 2n−2

−1 2n−2

−1 2n−2

−1 2n−2

0

··· .. .

−1 2n−2

−1 2n−2

.. .

.. .

···

λ

··· .. .

−1 2n−2

··· ··· ···

··· ··· ···

−1 2n−2 −1 2n−2 −1 2n−2

··· .. .

−1 2n−2

−1 2n−2

···

−1 2n−2

.. .

λ .. .

··· .. .

···

−1 2n−2

−1 2n−2

···

···

0

−1 2n−2

···

.. . −1 2n−2

.. . λ −1 2n−2

−1

2n−2

−1

2n−2

−1

2n−2

−1

2n−2

..

.

.

0

−1

2n−2

..

.

−1

2n−2

λ

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Minimum Equitable Dominating Randić Energy of a Graph

′ Rk′ = Rk − R3 , k = 4, 5, · · · , n and Rn+k = Rn+k − Rk+1 , k = 2, 3, · · · , n, we get

λ − 1

0

−1

2n−2

0

.

.

. |ρI−RED (Kn×2 )| =

0

0

.

..

0

0

0

−1 2n−2

−1 2n−2

λ−1

−1 2n−2

−1 2n−2

λ

−1 2n−2

−1 2n−2

0 .. .

−1 2n−2

0

−1 2n−2

.. .

−λ

λ+

−1 2n−2

−1 2n−2

−1 2n−2

···

−1 2n−2

0

··· .. .

0 .. .

−1 2n−2

···

1 2n−2

.. .

−λ

−1 2n−2

···

0

···

−1 2n−2

··· ···

−1 2n−2

··· .. .

0 .. .

−1 2n−2

···

0

.. .

1 2n−2

λ+

−1 2n−2

···

0 .. .

−λ .. .

0 .. .

··· .. .

0 .. .

λ .. .

··· .. .

0 .. .

0

0

0

···

0

0

···

λ

0

0

0

···

−λ

0

···

0

−1

2n−2

−1

2n−2

−1

2n−2

.

1

2n−2

0

..

.

0

λ

0 .. .

C3′ = C3 + C4 + · · · + Cn + Cn+1 + · · · + C2n and Ck′ = Ck + Cn+(k−1) , k = 4, 5, · · · , n + 1.

We get

λ − 1

0

−1

2n−2

0

.

.

. |ρI−RED (Kn×2 )| =

0

0

.

..

0

0

0

−1

λ−1

−1

−1 2n−2

λ−

−1 n−1 −1 n−1

(2n−4) (2n−2)

−1 2n−2

−1 n−1

−1 2n−2

···

−1 n−1

0

··· .. .

0 .. .

−1 2n−2

···

−1 n−1 1 n−1

λ+

−1 n−1

···

−1 2n−2

···

−1 2n−2

··· ···

−1 2n−2

··· .. .

0 .. .

−1 2n−2

···

0

0 .. .

0 .. .

0

0

0

···

0 .. .

0 .. .

0 .. .

··· .. .

0 .. .

λ .. .

··· .. .

0 .. .

0

0

0

···

0

0

···

λ

0

0

0

···

0

0

···

0

.. .

λ+

.. .

1 n−1

|ρI − RED (Kn×2 )| = SpecRED (Kn×2 ) =

ρ 

(ρ − 1) ρ +

−1 n−1

0

 n−2 n−1

1 n−1

n−2

2

ρ −

1

√ (2n−3)− 4n−3 2(n−1)

1

1

2n − 3 n−1

n−3 ρ+ n−1  √

(2n−3)+ 4n−3 2(n−1) 

1

−1

2n−2

−1

2n−2

.

1

2n−2

0

..

.

0

λ

0 .. .

The characteristic polynomial n−1

−1

2n−2

,

.

The minimum equitable dominating Randić energy of Kn×2 is REED (Kn×2 ) = where n ≥ 3.

4n − 6 , n−1

2

108

Rajendra P. and R.Rangarajan

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